Hi Andy, thanks for the very speedy response. 
I promise I’ve read everything I could find before posting, from (definitely) the docs, to SO, obscure GitHub issues from other clients/libraries, etc. Understood that this is at the account level.
I had hoped it was a 15-minute window, but no luck. As mentioned, I tried a command-line client, twurl, and pointed it at the same endpoint with a single user id:
twurl -d 'screen_name=fakename' -X POST /1.1/friendships/destroy.json
and received the same error in a JSON payload:
{"errors":[{"code":34,"message":"Sorry, that page does not exist."}]}
I thought the library I was using throttled requests to prevent such issues, but perhaps not and it triggered this response, even with a small set of requests. (As I’m assuming the real problems occur with people firing thousands of requests at the API!)
Also, a similar request with the twurl client to a different resource endpoint works fine (so it’s not the library or my account–it’s only one endpoint):
twurl -d 'id=12345' -X POST /1.1/direct_messages/destroy.json
That works, and you can see the only difference is the resource name.
Maybe it “heals” itself tomorrow, outside of 24 hours.
Incidentally, I suspect this is what occurred with a number of the issues I read for different libraries/clients on GitHub. The pattern would usually be: “P1: I’m getting this (34) error. P2: Your fault, you must be using the old API or an incorrect URL. P1 (next day): Works now! (issue gets closed)”
Or, the maintainer couldn’t reproduce and assumes it was an error by the user in constructing a request, when this same rate limiting issue (assuming that’s what it is) causes an error for one person (because they’ve hit a limit) but not another.